Fixed-Parameter Tractability and Completeness I: Basic Results
SIAM Journal on Computing
Journal of Combinatorial Theory Series B
Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics
CP '02 Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming
Graphical Models for Game Theory
UAI '01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence
Pure Nash equilibria: hard and easy games
Proceedings of the 9th conference on Theoretical aspects of rationality and knowledge
Computing Nash equilibria of action-graph games
UAI '04 Proceedings of the 20th conference on Uncertainty in artificial intelligence
The complexity of counting homomorphisms seen from the other side
Theoretical Computer Science
DAG-width: connectivity measure for directed graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Computing pure nash equilibria in graphical games via markov random fields
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
The complexity of homomorphism and constraint satisfaction problems seen from the other side
Journal of the ACM (JACM)
Digraph measures: Kelly decompositions, games, and orderings
Theoretical Computer Science
A polynomial-time algorithm for action-graph games
AAAI'06 Proceedings of the 21st national conference on Artificial intelligence - Volume 1
Symmetries and the complexity of pure Nash equilibrium
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
The complexity of games on highly regular graphs
ESA'05 Proceedings of the 13th annual European conference on Algorithms
D-Width: a more natural measure for directed tree width
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Parameterized Complexity
A study on the stability and efficiency of graphical games with unbounded treewidth
Proceedings of the 2013 international conference on Autonomous agents and multi-agent systems
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We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NP-complete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for the case of bounded in-degree graphs, thereby resolving the gap between existing hardness and tractability results. In particular, we analyze the complexity of PURE-GG(C, -), the problem of deciding the existence of pure Nash equilibria in graphical games whose underlying graphs are restricted to class C. We prove that, under reasonable complexity theoretic assumptions, for every recursively enumerable class C of directed graphs with bounded in-degree, PURE-GG(C, -) is in polynomial time if and only if the reduced graphs (the graphs resulting from iterated removal of sinks) of C have bounded treewidth. We also give a characterization for PURE-CHG(C, -), the problem of deciding the existence of pure Nash equilibria in colored hypergraphical games, a game representation that can express the additional structure that some of the players have identical local utility functions. We show that the tractable classes of bounded-arity colored hypergraphical games are precisely those whose reduced graphs have bounded treewidth modulo homomorphic equivalence. Our proofs make novel use of Grohe's characterization of the complexity of homomorphism problems.